Classical phase space and statistical mechanics of identical particles (original) (raw)

2001, Physical Review E

https://doi.org/10.1103/PHYSREVE.63.026102

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Abstract

Starting from the quantum theory of identical particles, we show how to define a classical mechanics that retains information about the quantum statistics. We consider two examples of relevance for the quantum Hall effect: identical particles in the lowest Landau level, and vortices in the Chern-Simons Ginzburg-Landau model. In both cases the resulting classical statistical mechanics is shown to be a nontrivial classical limit of Haldane's exclusion statistics.

Quantum mechanics from classical statistics

2009

Quantum mechanics can emerge from classical statistics. A typical quantum system describes an isolated subsystem of a classical statistical ensemble with infinitely many classical states. The state of this subsystem can be characterized by only a few probabilistic observables. Their expectation values define a density matrix if they obey a "purity constraint". Then all the usual laws of quantum mechanics follow, including Heisenberg's uncertainty relation, entanglement and a violation of Bell's inequalities. No concepts beyond classical statistics are needed for quantum physics -the differences are only apparent and result from the particularities of those classical statistical systems which admit a quantum mechanical description. Born's rule for quantum mechanical probabilities follows from the probability concept for a classical statistical ensemble. In particular, we show how the noncommuting properties of quantum operators are associated to the use of conditional probabilities within the classical system, and how a unitary time evolution reflects the isolation of the subsystem. As an illustration, we discuss a classical statistical implementation of a quantum computer.

Statistics, Symmetry, and the Conventionality of Indistinguishability in Quantum Mechanics

Foundations of Phyiscs, 2000

The question to be addressed is, In what sense and to what extent do quantum statistics for, and the standard formal quantum-mechanical description of, systems of many identical particles entail that identical quantum particles are indistinguishable? This paper argues that whether or not we consider identical quantum particles as indistinguishable is a matter of theory choice underdetermined by logic and experiment.

Statistical approach to quantum mechanics I: General nonrelativistic theory

In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schr\"odinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the NNN-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum NNN-vector replaced by −ihbar-i\hbar−ihbar times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. ...

On the explanation for quantum statistics

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2006

The concept of classical indistinguishability is analyzed and defended against a number of well-known criticisms, with particular attention to the Gibbs'paradox. Granted that it is as much at home in classical as in quantum statistical mechanics, the question arises as to why indistinguishability, in quantum mechanics but not in classical mechanics, forces a change in statistics. The answer, illustrated with simple examples, is that the equilibrium measure on classical phase space is continuous, whilst on Hilbert space it is discrete. The relevance of names, or equivalently, properties stable in time that can be used as names, is also discussed.

ELECTROMAGNETIC EXCITATIONS OF A n QUANTUM HALL DROPLETS

International Journal of Modern Physics A, 2010

The classical description of A n internal degrees of freedom is given by making use of the Fock-Bargmann analytical realization. The symplectic deformation of phase space, including the internal degrees of freedom, is discussed. We show that the Moser's lemma provides a mapping to eliminate the fluctuations of the symplectic structure, which become encoded in the Hamiltonian of the system. We discuss the relation between Moser and Seiberg-Witten maps. One physics applications of this result is the electromagnetic excitation of a large collection of particles, obeying the generalized A n statistics, living in the complex projective space CP k with U (1) background magnetic field. We explicitly calculate the bulk and edge actions. Some particular symplectic deformations are also considered. Quantum Hall effect in higher dimensions has intensively been investigated in the last decade from different point of views . This yielded interesting results such as the bosonization [2, 11-13] that has been achieved by making use of the incompressible Hall droplet picture. In this framework, the edge excitations of a quantum Hall droplet are described by a generalized Wess-Zumino-Witten action. Recently, the electromagnetic excitations of a quantum Hall droplet was discussed by Karabali [11] and Nair for the Landau systems in the complex projective space CP k . In fact, the corresponding bulk and edge actions were derived . Interestingly, it was shown that the bulk contribution coincides with the (2k + 1)-dimensional Chern-Simons action .

Quantum field theory from classical statistics

2011

An Ising-type classical statistical model is shown to describe quantum fermions. For a suitable time-evolution law for the probability distribution of the Ising-spins our model describes a quantum field theory for Dirac spinors in external electromagnetic fields, corresponding to a mean field approximation to quantum electrodynamics. All quantum features for the motion of an arbitrary number of electrons and positrons, including the characteristic interference effects for two-fermion states, are described by the classical statistical model. For one-particle states in the non-relativistic approximation we derive the Schrödinger equation for a particle in a potential from the time evolution law for the probability distribution of the Ising-spins. Thus all characteristic quantum features, as interference in a double slit experiment, tunneling or discrete energy levels for stationary states, are derived from a classical statistical ensemble. Concerning the particle-wave-duality of quantum mechanics, the discreteness of particles is traced back to the discreteness of occupation numbers or Ising-spins, while the continuity of the wave function reflects the continuity of the probability distribution for the Ising-spins.

Quantum Mechanics as a Classical Theory IX: The Formation of Operators and Quantum Phase-Space Densities

1995

In our previous papers we were interested in making a reconstruction of quantum mechanics according to classical mechanics. In this paper we suspend this program for a while and turn our attention to a theme in the frontier of quantum mechanics itself---that is, the formation of operators. We then investigate all the subtleties involved in forming operators from their classical counterparts. We show, using the formalism of quantum phase-space distributions, that our formation method, which is equivalent to Weyl's rule, gives the correct answer. Since this method implies that eigenstates are not dispersion-free we argue for modifications in the orthodox view. Many properties of the quantum phase-space distributions are also investigated and discussed in the realm of our classical approach. We then strengthen the conclusions of our previous papers that quantum mechanics is merely an extremely good approximation of classical statistical mechanics performed upon the configuration s...

Quantum mechanics as a classical theory; 9, the formation of operators and quantum phase-space densities

1995

Phase-Consistent Single-Particle Quantum Mechanics: Statistical Emergence of Born's Rule

A Phase-Consistent Single-Particle Quantum Mechanics framework is presented, proposing a "Phase Consistency Criterion" that treats the global phase of the quantum state vector as a hidden variable linking particle and wave behaviors. It also establishes a "quantum equivalent of Hamilton's principle of stationary action," which, when applied within Feynman's path integral formulation, determines the trajectory of a quantum particle. Together, these principles provide a physical explanation for wavefunction collapse, resolve the measurement problem, and describe the quantum-to-classical transition, offering a mechanistic foundation for orthodox quantum mechanics. In the case of large particle ensembles, the global phase acts as a statistical parameter, leading to a natural derivation of Born's rule.

Phase space reduction and vortex statistics: An anyon quantization ambiguity

Physical Review D, 1994

We examine the quantization of the motion of two charged vortices in a Ginzburg-Landau theory for the fractional quantum Hall effect recently proposed by the first two authors. The system has two second-class constraints which can be implemented either in the reduced phase space or Dirac-Gupta-Bleuler formalism. Using the intrinsic formulation of statistics, we show that these two ways of implementing the constraints are inequivalent unless the vortices are quantized with conventional statistics; either fermionic or bosonic.

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References (26)

M.G.C. Laidlaw and C.M.DeWitt, Feynman functional integrals for sys- tems of indistinguishable particles, Phys. Rev. D 3 (1971) 1375.
J.M. Leinaas and J. Myrheim, On the Theory of Identical Particles, Nuovo Cimento 37B (1977) 1.
G.A. Goldin, R. Menikoff and D.H. Sharp, Particle Statistics from In- duced Representations of a Local Current Group, J. Math. Phys. 21 (1980) 650 ; Representations of a Local Current Algebra in Nonsimply Connected Space and the Aharonov-Bohm Effect, J. Math. Phys. 22 (1981) 1664;
F. Wilczek, Magnetic flux, angular momentum and statistics, Phys. Rev. Lett. 48 (1982) 1144 ; Quantum mechanics of fractional-statistics particles, Phys. Rev. Lett. 49 (1982) 957.
F.D.M. Haldane, Fractional statistics in arbitrary dimensions -A gen- eralization of the Pauli principle, Phys. Rev. Lett. 67 (1991) 937.
T.M. Samols, Vortex Scattering, Commun. Math. Phys. 145 (1992) 149.
N.S. Manton, Statistical mechanics of vortices, Nucl. Phys. B 400 [FS] (1993) 624.
M.V. Berry, Quantal phase factors acompanying adiabatic changes, Proc. Roy. Soc. (London) A 392 (1984) 45.
R. Jackiw and L. Faddeev Hamiltonian reduction of unconstrained and constrained systems, Phys. Rev. Lett. 60 (1988) 1692.
J.P. Provost and G. Vallee, Riemannian Structure on Manifolds of Quantum States, Commun. Math. Phys. 76 (1980) 289.
A. Perelemov, Generalized Coherent States and Their Applications Springer-Verlag 1986.
See e.g., A. Karlhede and E. Westerberg, Anyons in a magnetic field, Int. Jour. Mod. Phys. B 6 (1992) 1595.
H. Kjønsberg and J.M. Leinaas, On the anyon description of the Laugh- lin hole states, Int. Jour. Mod. Phys. A 12 (1997) 1975.
H. Kjønsberg and J. Myrheim, Numerical study of charge and statistics of Laughlin quasi particles, Int. Jour. Mod. Phys. A 14 (1999) 537.
F. D. M. Haldane, Fractional Quantization of the Hall Effect: A Hierar- chy of Incompressible Quantum Fluid States, Phys. Rev. Lett. 51 (1983) 605.
F.D.M. Haldane and E.H. Rezayi, Periodic Laughlin-Jastrow wave func- tions for the fractional quantized Hall effect , Phys. Rev. B 31 (1985) 2529.
N. S. Manton, S. M. Nasir, Volume of Vortex Moduli Spaces (partition functions on a Riemann surface), Commun. Math. Phys. 199 (1999) 591.
S. M, Girvin and A. H. MacDonald, Off-Diagonal Long-Range Order, Oblique Confinement, and the Fractional Quantum Hall Effect. Phys. Rev. Lett. 58, 1252 (1987);
S. C. Zhang, T. H. Hansson and S. A. Kivelson, Effective-field-theory model for the fractional quantum Hall effect, Phys. Rev. Lett. 62 (1989) 82.
N. S. Manton, "First order vortex dynamics," Annals Phys. 256, 114 (1997) [hep-th/9701027].
C.H. Taubes, Arbitrary N-Vortex Solutions to the First Order Ginzburg- Landau Equations, Commun. Math. Phys. 72 (1980) 277, 75 (1980) 207.
D.Arovas, R.Schriffer and F.Wilczek, Fractional Statistics and the Quan- tum Hall Effect, Phys. Rev. Lett. 53 (1984) 722
S.B. Isakov, Generalization of statistics for several species of identical particles, Mod. Phys. Lett. B 8 (1994) 319; Fractional statistics in one dimension: modeling by means of 1/x 2 in- teraction and statistical mechanics; Int. J. Mod. Phys. A 9 (1994) 2563.
A. Dasnières de Veigy and S. Ouvry, Equation of state of an anyon gas in a strong magnetic field, Phys. Rev. Lett. 72 (1994) 600.
Y.S. Wu, Statistical distribution for generalized ideal-gas of fractional- statistics particles, Phys. Rev. Lett. 73 (1994) 922.
A. Comtet, Y. Georgelin and S. Ouvry, Statistical aspects of the anyon model, J. Phys. A: Math. Gen. 22 (1989) 3917; K. Olaussen, On the har- monic oscillator regularization of partition function, Trondheim Univ. preprint No. 13 (1992), cond-mat/9207005.

Quantum particles from classical statistics

2009

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being "classical" or "quantum" ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross-fertilization between classical statistics and quantum physics.

Statistical mechanics of classical systems with distinguishable particles

2002

The properties of classical models of distinguishable particles are shown to be identical to those of a corresponding system of indistinguishable particles without the need for ad hoc corrections. An alternative to the usual definition of the entropy is proposed. The new definition in terms of the logarithm of the probability distribution of the thermodynamic variables is shown to be consistent with all desired properties of the entropy and the physical properties of thermodynamic systems. The factor of 1/N! in the entropy connected with Gibbs' Paradox is shown to arise naturally for both distinguishable and indistinguishable particles. These results have direct application to computer simulations of classical systems, which always use distinguishable particles. Such simulations should be compared directly to experiment (in the classical regime) without ''correcting'' them to account for indistinguishability.

Classical particles and order statistics

Physics Letters, 1989

It is shown that classical particles obey order statistics. This resolves the paradox of having to write in by hand the 1/Ni term in the N-particle partition function: This factor arises precisely because the particles are distinguishable rather than because there are Ni ways of permuting among themselves N particles which are physically indistinguishable and must be counted only once. Recently, the old and unsettled problem of whether dynamic properties, such as extensivity. The factor classical and quantum particles are distinguishable N! which must be introduced to achieve this, is noror not has been brought to the foreground. If quan-mally interpreted as indicating that classical partiturn statistics is a mere variation of classical statis-des are really indistinguishable.

Distinguishability or indistinguishability in classical and quantum statistics

Physics Letters, 1987

Contrary to a recent formulation ofclassical and quantum statistics in terms ofindistinguishable particles it is claimed that the notion of distinguishability can account for classical as well as for quantum statistics. The resulting non-local correlations are a manifestation ofa fluctuating stochastic metric which has been shown to produce the quantum potential.

General approach to quantum mechanics as a statistical theory

Physical review, 2019

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function-the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics-extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phasespace quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space-putting it for the first time on a truly equal footing to Schrödinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

The Phase Space Elementary Cell in Classical and Generalized Statistics

Entropy, 2013

In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementary-cell volume of a system of non-interacting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated non-Boltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h 3 , at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger) phase-space volume described by non-extensive generalized statistics.

Statistical mechanics of generally covariant quantum theories: A Boltzmann-like approach

We study the possibility of applying statistical mechanics to generally covariant quantum theories with a vanishing Hamiltonian. We show that (under certain appropriate conditions) this makes sense, in spite of the absence of a notion of energy and external time. We consider a composite system formed by a large number of identical components, and apply Boltzmann's ideas and the fundamental postulates of ordinary statistical physics. The thermodynamical parameters are determined by the properties of the thermalizing interaction. We apply these ideas to a simple example, in which the component system has one physical degree of freedom and mimics the constraint algebra of general relativity.

Quantum Dynamics Arising from Statistical Axioms

2010

We investigate the dynamics of pairs of Fermions and Bosons released from a box and find that their populations have unique generic properties ensuing from the axioms of quantum statistics and symmetries. These depend neither on the specific equations of wave function propagation, such as Schr\"odinger, Klein-Gordon, Dirac, nor on the specific potential involved. One surprising finding is that after releasing the pairs, there are always more Boson than Fermion pairs outside the box. Moreover, if the initial wave functions have the same symmetry (odd or even), then there is a higher chance for a Boson than a Fermion pair to escape from the trap in opposite directions, as if they repel each other. We calculate the wave functions exactly, numerically, and asymptotically for short time and demonstrate these generic results in the specific case of particles released from an infinite well.

Level statistics for quantum Hall systems

Low Temperature Physics, 2005

Level statistics for two classes of disordered systems at criticality are analyzed in terms of different realizations of the Chalker-Coddington network model. These include: 1) Re-examination of the standard U() 1 model describing dynamics of electrons on the lowest Landau level in the quantum Hall effect, where it is shown that after proper local unfolding the nearest-neighbor spacing distribution (NNSD) at the critical energy follows the Wigner surmise for Gaussian unitary ensembles (GUE). 2) Quasi-particles in disordered superconductors with broken time reversal and spin rotation invariance (in the language of random matrix theory this system is a representative of symmetry class D in the classification scheme of Altland and Zirnbauer). Here again the NNSD obeys the Wigner surmise for GUE, reflecting therefore only «basic» discrete symmetries of the system (time reversal violation) and ignoring particle-hole symmetries and other finer details (criticality). In the localized regime level repulsion is suppressed.

Statistical mechanics of quantum-classical systems

The Journal of Chemical Physics, 2001

The statistical mechanics of systems whose evolution is governed by mixed quantum-classical dynamics is investigated. The algebraic properties of the quantum-classical time evolution of operators and of the density matrix are examined and compared to those of full quantum mechanics. The equilibrium density matrix that appears in this formulation is stationary under the dynamics and a method for its calculation is presented. The response of a quantum-classical system to an external force which is applied from the distant past when the system is in equilibrium is determined. The structure of the resulting equilibrium time correlation function is examined and the quantum-classical limits of equivalent quantum time correlation functions are derived. The results provide a framework for the computation of equilibrium time correlation functions for mixed quantum-classical systems.

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Classical phase space and statistical mechanics of identical particles (original) (raw)

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